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(Symbol β2 or α4.) A descriptive measure of a random variable in terms of the flatness of its probability distribution.

It is defined as follows:
where μ4 is the fourth (statistical) moment about the mean and σ2 the variance. For the normal distribution, β2 = 3; cases for which β2 > 3 indicate distributions that are more outlier-prone than the normal (Gaussian) distribution, while those for which β2 < 3 indicate distributions that are less outlier-prone than the normal. In particular, the rectangular distribution f(x) = 1 (0 < x < 1) has β2 = 1.8. The terms leptokurtic, mesokurtic, and platykurtic refer to curves for which the values of β2 are, respectively, greater than 3, equal to 3, and less than 3. Excess is a relative expression for kurtosis, and the coefficient of excess γ2 is defined as β2 − 3.

Term edited 16 October 2017.

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